I started diving last fall and can impart my wisdom, sparse as it might be, to someone looking to get into the sport.
I bought my first mask, snorkel and fins in a set from Oceanic with the purpose of using them for snorkeling. I had considered diving, but I wasn't thinking about it when I bought them. Turns out that the fins are good (large split fins, open heel), but the mask and snorkel are now relegated to my backup snorkel and my last resort mask.
For the fins, you should probably see if you can borrow or rent some to try out to see what you like. Here in Okinawa, shore diving is the norm so booties with open-heel fins are the only option because you are NOT walking on coral with bare feet -- can you say "bloody water"? The booties are also nice because they can double as shoes, and they protect your feet and ankles from things such as sea cucumbers that are pissed that you touched them. Fins come in a variety of styles, and you need to figure out what you want. The big rage is split fins, in particular Scubapro TwinJets (at least around here -- special ops guys use them, seriously!), but they cost around $140. Some people don't like them. Others prefer stiffer, non-split styles that are apparently better in a current but somewhat more tiring. I am certainly no expert, but the point is you need to figure out what you like.
For the snorkel, don't go too cheap. While you may not use it all that much for boat diving, even a short swim on the surface with a water-conduit snorkel is a pain. Look for one with a baffle of some sort on the intake and a trap with a blowhole on the bottom.
On the mask, this is one of the most important pieces of equipment which some people overlook at first. I did! First and foremost, decide whether or not you want to use a purge valve. Some people insist that it's an unnecessary addition that only adds another point of failure to your equipment. Others say that it's fine and adds a significant level of ease and comfort. I used to be in the second group, but now I'm in the first. I had a purge valve mask and used it for four dives without any problems. It was very easy to use, and I loved it. Until two dives ago when it failed on me, and I couldn't go below 20 feet without the mask flooding (valve was shot and water pressure forced past it at depth). It ruined my dive. So, I bought a non-purge and dove with that on the last one -- great mask (U.S. Divers Hawaii 3). Also, think about if you want side panels or not. The volume of the mask increases with side panels, which makes equalization and clearing slightly more difficult, but the addition to your local awareness is very noticeable and worth it to me. I have noticed that the people I dive with who do not have side panels tend to be oblivious of the people right next to them and are more likely to kick you or swim into you accidentally because they simply can't see without moving their entire head.
Again here in Okinawa, we are very fortunate to have numerous military-sanctioned dive shops which are forced to operate with low profit margins because of the vendor rules on military installations. So, our equipment prices are awesome, and they are from authorized dealers, which means that the equipment manufacturers will honor the warranties. Not so with the online dealers. Their prices are great, but they are NOT authorized dealers and therefore basically have no warranties (I learned this from a scuba message board, not experienced personally). That's fine for masks and snorkels and such, but beware if you buy regulators, BCs, etc. Plus, as is mentioned above, it's better to establish a relationship with a local dive shop. You can then learn to trust their opinions about dive sites, equipment, techniques, etc, and even better, they'll stay open so you can keep going there. It's equivalent to buying from a fly-by-night computer store or shopping at Dell. Sure, the Dells are more expensive, but you can trust the equipment and the service much more than the here-today-gone-tomorrow store.
Can't help you on the weight belts since I have integrated weights in my BC. The dive shop here let us rent weights for the class, and I bought my gear prior to my second set of open water dives (couldn't stand the rental stuff!). Also consider buying a smallish dive light. I have one that I carry all the time simply because you can then look into crevices that are constantly in shadow even on the brightest days. I've spotted some critters that I otherwise would not have seen with that light. Don't worry about rechargeables unless you're planning on doing a lot of night diving down the road (and they are stupidly expensive!).
Don't forget about a watch, too. While you can always buy a dive computer somewhere down the line, a good dive watch is essential to track your time underwater and even with a computer serves as a backup in case of failure. I have a Timex Reefgear watch that worked fairly well until the battery stopped prematurely. I've since bought a Seiko, but it's a little more pricey ($250 vs. $30 for the Timex!).
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